Recently I heard the
theorem that any (in both directions) infinite sequence of real numbers xn
such that for all n

|xn| = xn-1 + xn+1

(0)

has a period of
length 9. Here is my proof.

p

p + r

(≥
0)

r

(≥
0)

-p

(≤
0)

p - r

(≥
0)

2∙p -
r

(≥
0)

p

(≥
0)

r - p

(≤
0)

- r

(≤
0)

p

(≥
0)

p + r

From
(0) we conclude (i) that the sequence contains a nonnegative
element, (ii) that one of its neighbours is nonnegative, and (iii)
that at least one of the two elements adjacent to a pair of
nonnegative neighbours is nonnegative. More precisely: the
sequence contains in some direction a triple of adjacent elements
of the form (p, p+r, r) with 0 ≤r≤p. To the left we have extended the sequence with another 8
elements. From (0) we further conclude that the whole sequence is
determined by a pair of adjacent values; hence, the repetition of
the pair (p, p+r) at distance 9 proves the
theorem. [The above deserves recording for its lack of case
analyses.]